Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

Okon, Ituen; Onate, Clement; Omugbe, Ekwevugbe; Okorie, Uduakobong; Antia, Akaninyene; Onyeaju, Michael; Wen-Li, Chen; Araujo, Judith

doi:https://doi.org/10.1155/2022/5178247

Advances in High Energy Physics

On this page

Abstract Introduction Discussion Conclusion Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5178247 | https://doi.org/10.1155/2022/5178247

Approximate Solutions, Thermal Properties, and Superstatistics Solutions to Schrödinger Equation

Ituen Okon,¹Clement Onate,²Ekwevugbe Omugbe,³Uduakobong Okorie,⁴Akaninyene Antia,¹Michael Onyeaju,⁵Chen Wen-Li,⁶and Judith Araujo⁷

Academic Editor: Mariana Frank

Received16 Oct 2021

Accepted17 Feb 2022

Published07 Mar 2022

Abstract

In this work, we apply the parametric Nikiforov-Uvarov method to obtain eigensolutions and total normalized wave function of Schrödinger equation expressed in terms of Jacobi polynomial using Coulomb plus Screened Exponential Hyperbolic Potential (CPSEHP), where we obtained the probability density plots for the proposed potential for various orbital angular quantum number, as well as some special cases (Hellmann and Yukawa potential). The proposed potential is best suitable for smaller values of the screening parameter . The resulting energy eigenvalue is presented in a close form and extended to study thermal properties and superstatistics expressed in terms of partition function and other thermodynamic properties such as vibrational mean energy , vibrational specific heat capacity , vibrational entropy , and vibrational free energy . Using the resulting energy equation and with the help of Matlab software, the numerical bound state solutions were obtained for various values of the screening parameter () as well as different expectation values via Hellmann-Feynman Theorem (HFT). The trend of the partition function and other thermodynamic properties obtained for both thermal properties and superstatistics were in excellent agreement with the existing literatures. Due to the analytical mathematical complexities, the superstatistics and thermal properties were evaluated using Mathematica 10.0 version software. The proposed potential model reduces to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases.

1. Introduction

The approximate analytical solutions of one-dimensional radial Schrödinger equation with a multiple potential function have been studied using a suitable approximation scheme to the centrifugal term within the frame work of the parametric Nikiforov-Uvarov method [1]. The solutions to the wave equations in quantum mechanics and applied physics play a crucial role in understanding the importance of physical systems [2]. The two most important parts in studying Schrödinger equations are the total wave function and energy eigenvalues [3]. The analytic solutions of wave equations for some physical potentials are possible for . For , special approximation schemes like the Greene-Aldrich and Pekeris approximations are employed to deal with the centrifugal barrier in order to obtain approximate bound state solutions [4–6]. The Greene-Aldrich approximation scheme is mostly applicable for short range potentials [7]. Eigensolutions for both relativistic and nonrelativistic wave equations have been studied with different methods which include the following: Exact quantisation, WKB, Nikiforov-Uvarov method (NU), Laplace transform technique, asymptotic iteration method, proper quantisation, supersymmetric quantum mechanics approach, vibrational approach, formula method, factorisation method, and Shifted 1/N-expansion method [8–13]. Bound state solutions obtained from the Schrödinger equation has practical applications in investigating tunnelling rate of quantum mechanical systems [14] and mass spectra of quarkonia systems [15–19]. Among other goals achieved in this research article is to apply the Hellmann-Feynman Theorem (HFT) to eigenequation of the Schrödinger wave equations to obtain expectation values of , , , and analytically. The Hellmann-Feynman Theorem gives an insight about chemical bonding and other forces existing among atoms of molecules [20–25]. To engage HFT in calculating the expectation values, one needs to promote the fixed parameter which appears in the Hamiltonian to be a continuous variable in order to ease the mathematical purpose of taking the derivative [26]. Similarly, the application of Hellmann-Feynman Theorem provides a less mathematical approach of obtaining expectation values of a quantum mechanical systems [27, 28]. Some of the potential models considered within the framework of relativistic and nonrelativistic wave equations are Hulthen-Yukawa Inversely quadratic potential [29], noncentral Inversely quadratic potential [30], Modified Hylleraas potential [31], Yukawa, Hulthen, Eckart, Deng-Fan, Pseudoharmonic, Kratzer, Woods-Saxon, double ring shape, Coulomb, Tietz–Wei, Tietz-Hua, Deng-Fan, Manning-Rosen, trigonometric Rosen-Morse, hyperbolic scalar, and vector potential and exponential type potentials among others [32–50]. Coulomb, hyperbolic, and screened exponential type potentials have been of interest to researchers in recent times because of their enormous applications in both chemical and physical sciences. In view of this, Parmar [51] studied ultrageneralized exponential hyperbolic potential where he obtained energy eigenvalues, unnormalized wave function and the partition function. This potential reduces to Yukawa potential, Screened cosine Kratzer potential, Manning-Rosen potential, Hulthen plus Inversely quadratic exponential Mie-type potential, and many others. Diaf et al. [52], in their studies, obtained eigensolutions to the Schrödinger equation with trigonometric Inversely quadratic plus Coulombic hyperbolic potential where they obtained energy eigenvalue and normalized wave function using the Nikiforov-Uvarov method. Onate [53] examined bound state solutions of the Schrödinger equation with second Pöschl-Teller-like potential where he obtained vibrational partition function, mean energy, vibrational specific heat capacity, and mean free energy. In that work, the Pöschl-Teller-like potential was expressed in a hyperbolic form. The practical application of energy eigenvalue of Schrödinger equation in investigating the partition function, thermodynamic properties, and superstatistics arouses the interest of many researchers. Recently, Okon et al. [54] obtained the thermodynamic properties and bound state solutions of the Schrödinger equation using Mobius square plus screened Kratzer potential for two diatomic systems (carbon(II) oxide and scandium fluoride) within the framework of the Nikiforov-Uvarov method. Their results were in agreement to semiclassical WKB among others. They presented energy eigenvalue in a close form in order to obtain partition function and other thermodynamic properties. Omugbe et al. [55] recently studied the unified treatment of the nonrelativistic bound state solutions, thermodynamic properties, and expectation values of exponential-type potentials where they obtained the thermodynamic properties within the framework of semiclassical WKB approach. The authors studied the special cases of the potential as Eckart, Manning-Rosen, and Hulthen potentials. Besides, Oyewumi et al. [56] studied the thermodynamic properties and the approximate solutions of the Schrödinger equation with shifted Deng-Fan potential model within the framework of asymptotic Iteration method where they apply Pekeris-type approximation to centrifugal term to obtain rotational-vibrational energy eigenvalues for selected diatomic systems. A lot of researches have been carried out by Ikot et al. These can be seen in Refs. [57–60]. Also, Boumali and Hassanabadi [61] studied thermal properties of a two-dimensional Dirac oscillator under an external magnetic field where they obtained relativistic spin-1\2 fermions subject to Dirac oscillator coupling and a constant magnetic field in both commutative and noncommutative spaces.

In this work, we propose a novel potential called Coulomb plus Screened Exponential Hyperbolic Potential to study bound state solutions, expectation values, superstatistics, and thermal properties within the framework of the parametric Nikiforov-Uvarov method [62]. This article is divided into 9 sections. The introduction is given in Section 1. The parametric Nikiforov-Uvarov method is presented in Section 2. The solutions of the radial Schrödinger equation are presented in Section 3. The application of Hellmann-Feynman Theorem to obtain expectation values is presented in Section 4. The thermodynamic properties and superstatistics formulations are presented in Sections 5 and 6, respectively. Numerical results and discussion are presented Sections 7 and 8, respectively, and the article is concluded in Section 9.

The propose Coulomb plus Screened Hyperbolic Exponential Potential (CPSHEP) is given as where and are the potential depths, is a real constant parameter, and is the adjustable screening parameter. The Pekeris-like approximation to the centrifugal term is given as

The graph of Pekeris approximation to centrifugal term is given in Figure 1.

2. Parametric Nikiforov-Uvarov (NU) Method

The NU method is based on reducing second order linear differential equation to a generalized equation of hypergeometric type and provides exact solutions in terms of special orthogonal functions like Jacobi and Laguerre as well as corresponding energy eigenvalues [63–70]. The reference equation for parametric NU method according to Tezcan and Sever [71] is given as

The condition for energy equation is given as [70].

The total wave function is given as

The parametric constants are obtained as follows:

3. The Radial Solution of Schrödinger Wave Equation

The radial Schrödinger wave equation with the centrifugal term is given as

Equation (7) can only be solved analytically to obtain exact solution if the angular orbital quantum number . However, for , equation (7) can only be solve by using the approximations in (2) to the centrifugal term. Substituting equation (1) into (7) gives

By substituting equation (2) into (8) gives the following equation:

By defining and with some simple algebraic simplification, equation (9) can be presented in the form where

Comparing equation (10) to (3), the following polynomials were obtain:

Using equation (6), other parametric constants are obtained as follows:

Using equations (4), (12), and (13) with much algebraic simplification, the energy eigenvalue for the proposed potential is given as

Using equation (5), the total unnormalized wave function is given as

To obtain the normalization constant of equation (15), we employ the normalization condition where

The wave function is assumed to be in bound at and .

Equation (15) reduces to

Let such that the boundary of integration of equation (19) changes from to . Then, equation (19) reduces to

Using the standard integral,

Let . Then, using equation (20), the normalization constant can be obtained as

Hence, the total normalized wave function is given as

4. Expectation Values Using Hellmann-Feynman Theorem

In this section, some expectation values are obtain using Hellmann-Feynman Theorem (HFT). According to the Hellmann-Feynman Theorem, the Hamiltonian for a particular quantum mechanical system is expressed as a function of some parameters . Let and be the eigenvalues and eigenfunction of the Hamiltonian. Then,

For the purpose of clarity, the Hamiltonian for the propose potential using HFT is

4.1. Expectation Value for

To obtain the expectation value for , we set to have where

4.2. Expectation Value for

Setting , then, the expectation value for becomes

4.3. Expectation Value for

Setting , then,

4.4. Expectation Value for

The relationship between and is given as , therefore

5. Thermodynamic Properties

In this section, we present the thermodynamic properties for the potential model. The thermodynamic properties of quantum systems can be obtained from the exact partition function given by where is an upper bound of the vibrational quantum number obtained from the numerical solution of , given as , , where and are Boltzmann constant and absolute temperature, respectively. In the classical limit, the summation in equation (31) can be replaced with an integral:

In order to obtain the partition function, the energy equation (14) can be presented in a close and compact form as

Equation (33) can further be simplified to where and equation (34) can be represented in the form where (i)Partition function is obtained as follows:

Substituting equation (36) into equation (32) taking note of changes in the integration boundaries using equation (37) gives the partition function as

Using Mathematica 10.0 version, the partition function of equation (38) is given as (ii)Vibrational mean energy is given as(iii)Vibrational entropy is given as(iv)Vibrational free energy is given as(v)Vibrational specific heat capacity is given aswhere

6. Superstatistics Formulation

Superstatistics is the superposition of two different statistics which is applicable to driven nonequilibrium systems to statistical intensive parameter () fluctuation [72]. This intensive parameter which undergoes spatiotemporal fluctuations includes chemical potential and energy fluctuation which is basically describe in terms of effective Boltzmann factor [73]. According to Edet et.al [74], the effective Boltzmann factor is given as where is the Dirac delta function. However, the generalized Boltzmann factor expressed in terms of deformation parameter is given as

The partition function for superstatistics formalism is then given as

Substituting equation (34) into equation (46) gives the generalized Boltzmann factor equation as

Using equation (47), the superstatistics partition function equation is given as

Using Mathematica 10.0 version, the partition obtain from equation (47) is

We use the same procedure of thermodynamics (Section 5) to obtain superstatistics vibrational mean energy (), vibrational specific heat capacity (), vibrational entropy (), and vibrational free energy () from the partition equation (41). However, this solution is not included in the article because of the lengthy and bulky analytical equations.

7. Numerical Results

Using Matlab 10.0 version, the numerical bound state solutions for the proposed potential were calculated using (14) for different quantum state. Also, using equations (18), (20), (21), and (22), the expectation values for , , , and , respectively, were calculated as shown in Tables 1–5.

7.1. Special Cases

(a)Hellmann potential: substituting into equation (1), then, the potential reduces to Hellmann potential

The required energy equation is

The result of equation (52) agrees excellently with Ref. [75]. (b)Yukawa potential: if , then, equation (1) reduces to Yukawa potential

The corresponding energy eigenequation is

Equation (54) is consistent with result obtain in Ref. [76] in order to prove the mathematical accuracy of our analytical calculation. (c)Screened-hyperbolic inversely quadratic potential: substituting into equation (1), the potential reduces to screened-hyperbolic inversely quadratic potential

The resulting energy eigenvalue is (d)Coulomb potential: substituting into equation (53), then, the potential reduces to Coulomb potential

By substituting into equation (54), it gives the corresponding energy eigenvalue for Coulomb’s potential as

Equation (58) agrees we with result obtain in Ref [76].

8. Discussion

Figure 1 is the graph of Pekeris approximation against the screening parameter . The nature of the graph shows that the approximation is suitable for the proposed potential. Variation of the probability density against the internuclear separation at various quantum state for and , respectively, is shown in Figures 2(a) and 2(b), respectively. In Figure 2(a), the probability density curves produce thermal curves with regular peaks compacted close to the origin with uneven peaks at various internuclear distance. This curve shows that for orbital angular quantum number there is more concentration of the electron density at the origin for all the quantum state studied. The same situation is also observed for as shown in Figure 2(b). It can also be observed that at every value of the internuclear distance, the probability density for is higher than the probability density for . Figure 3 shows variation of the probability density against the internuclear separation at various quantum state for and for Hellmann potential presented in Figures 3(a) and 3(b), respectively. More concentration of the electron density is observed at the origin in both cases. It is also seen that the probability density obtained for is lower than the probability density obtained for In Figure 4, we presented the variation of the probability density against the internuclear separation at various quantum state for and for Yukawa potential as shown in Figures 4(a) and 4(b), respectively. Here, there are more concentration and localization of electron density at the origin for with uneven distribution of peak curves as shown in Figure 4(a), but this situation is not the same for as presented in Figure 4(b). The probability density at the second excited state for remains constant with uniform distribution of the density curves for all values of the internuclear separation.

(a)

(b)

(a)

(b)

(a)

(b)

In Figure 5, the variation of partition function for thermodynamic properties and superstatistics with the temperature parameter were observed. Here, the partition function increases nonlinearly with . However, in Figure 5(a), the partition function diverged as increases, but later converges for the superstatistics as shown in Figure 5(b). It can also be observed that the partition function converges as becomes positive in the superstatistics in Figure 5(b). In Figure 6, we presented the variation of vibrational mean energy against for thermodynamic properties and superstatistics, respectively, as shown in Figures 6(a) and 6(b). For the thermal property, the mean energy increases as goes up for all values of as shown in Figure 6(a), and at higher values of the mean energy for various tends to converge. The superstatistics mean energy rises as the temperature of the system decreases as shown in Figure 6(b). However, at a certain absolute temperature, the superstatistics mean energy increases vertically for various values of the deformed parameter (). The variation is opposite in Figure 7 which is the variation of the heat capacity against for different values of and deformed parameter is shown in Figures 7(a) and 7(b), respectively. In each case, the heat capacity decreases monotonically with an increasing for the nonsuperstatistics (thermodynamic property). At zero value of , the heat capacity for various converged and diverges as increases gradually. For the superstatistics, the heat capacity for various deformed parameter rises while the temperature cools down. The specific heat capacity has a turning point when equals -150 as shown in Figure 7(b) while the specific heat capacity for the nonsuperstatistics has a maximum turning point at about 0.15 as shown in Figure 7(a). In Figure 8, the vibrational entropy decreases and diverged while the temperature of the system decreases ( increases) in the nonsuperstatistics as shown in Figure 8(a). This decrease is sharper for negative values of the entropy when is almost constant as shown in Figure 8(b). The superstatistics entropy varies inversely with (directly with temperature). This means that when the temperature of the system is raised, the disorderliness of the system also increases for every value of the deformed parameter. The entropy for the superstatistics converged as the temperature parameter tends to zero. In Figure 9, the variation of free energy against the is seen to be two different steps in the case of a nonsuperstatistics as shown in Figure 9(a). Here, between 0 and 60 values of β, the free energy increases steadily for the various values of but beyond this range; the free energy increases sharply at constant . For the superstatistics, the free energy increases monotonically as the temperature of the system reduces gradually. The free energy is always higher when the deformed parameter is increased as shown in Figure 9(b).

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

(a)

(b)

Table 1 is the numerical bound state solutions for CPSEHP. The numerical bound state solutions increases with an increase in quantum state, but decreases with an increase in the screening parameter. Tables 2–5 are expectation values for , , , and , respectively. Here, the numerical values in all cases decrease with an increase in the screening parameter.

9. Conclusion

In this work, we apply the parametric Nikiforov-Uvarov method to obtain the bound state solutions of Coulomb plus Screened Exponential Hyperbolic Potential. The resulting energy eigenvalues were presented in a close and compact form. The research work was extended to study thermal properties, superstatistics, and various expectation values. The proposed potential also reduced to Hellmann potential, Yukawa potential, Screened Hyperbolic potential, and Coulomb potential as special cases. The normalized wave function for the mother potential and that of the Hellmann potential are similar, but the normalized wave function of the Yukawa potential seams different. The trend of the thermodynamic and superstatistics curves is in agreement to the results of an existing literature. The results of the thermodynamic properties and superstatistics revealed that the effect of the temperature on the thermodynamic properties and the superstatistics are similar. Finally, this research work has practical applications in physical and chemical sciences.

Data Availability

The numerical data used for this work are generated using Matlab programme.

Disclosure

An earlier version of this article has been deposited in arxiv in Cornell University which is accessible at https://arxiv.org/abs/2110.09896.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The article processing charge will be funded by SCOAP3.

References

C. A. Onate, M. C. Onyeaju, A. Abolarinwa, and A. F. Lukman, “Analytical determination of theoretic quantities for multiple potential,” Scientific Reports, vol. 10, no. 1, 2020.
View at: Publisher Site | Google Scholar
E. Omugbe, O. E. Osafile, I. B. Okon, and M. C. Onyeaju, “Energy spectrum and the properties of Schoiberg potential using the WKB approximation approach,” Molecular Physics, vol. 119, 2021.
View at: Publisher Site | Google Scholar
C. A. Onate, “Approximate solutions of the non-relativistic Schrődinger equation with an interaction of Coulomb and Hulthѐn potentials,” SOP Transactions on Theoretical Physics, vol. 1, no. 2, pp. 118–127, 2014.
View at: Publisher Site | Google Scholar
W. C. Qiang and S. H. Dong, “Arbitrary ℓ-state solutions of the rotating Morse potential through the exact quantization rule method,” Physical Letters A, vol. 363, no. 3, pp. 169–176, 2007.
View at: Publisher Site | Google Scholar
C. Berkdemir and J. Han, “Any ℓ-state solutions of the Morse potential through the Pekeris approximation and Nikiforov-Uvarov method,” Chemical Physics Letters, vol. 409, no. 4-6, pp. 203–207, 2005.
View at: Publisher Site | Google Scholar
C. Pekeris, “The rotation-vibration coupling in diatomic molecules,” Physical Review, vol. 45, no. 2, pp. 98–103, 1934.
View at: Publisher Site | Google Scholar
R. L. Greene and C. Aldrich, “Variational wave functions for a screened Coulomb potential,” Physical Review A, vol. 14, no. 6, pp. 2363–2366, 1976.
View at: Publisher Site | Google Scholar
A. D. Antia, I. B. Okon, A. O. Akankpo, and J. B. Usanga, “Non-relativistic bound state solutions of modified quadratic Yukawa plus q-deformed Eckart potential,” Journal of Applied Mathematics and Physics, vol. 8, no. 4, pp. 660–671, 2020.
View at: Publisher Site | Google Scholar
I. B. Okon, E. Omugbe, A. D. Antia, C. A. Onate, L. E. Akpabio, and O. E. Osafile, “Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential,” Scientific Reports, vol. 11, no. 1, p. 892, 2021.
View at: Publisher Site | Google Scholar
A. D. Antia, I. B. Okon, E. B. Umoren, and C. N. Isonguyo, “Relativistic study of the spinless Salpeter equation with a modified Hylleraas potential,” Ukranian Journal of Physics, vol. 64, no. 1, pp. 27–33, 2019.
View at: Publisher Site | Google Scholar
I. B. Okon and O. O. Popoola, “Bound- state solution of Schrodinger equation with Hulthen plus generalized exponential coulomb potential using Nikiforov-Uvarov method,” International Journal of Recent Advances in Physics, vol. 4, no. 3, pp. 1–12, 2015.
View at: Publisher Site | Google Scholar
A. D. Antia, C. C. Eze, I. B. Okon, and L. E. Akpabio, “Relativistic studies of Dirac equation with a spin-orbit coupled Hulthen potential including a Coulomb-like tensor interaction,” Journal of Applied and Theoretical Physics Research, vol. 3, no. 2, pp. 1–8, 2019.
View at: Publisher Site | Google Scholar
I. B. Okon, O. Popoola, C. N. Isonguyo, and A. D. Antia, “Solutions of Schrödinger and Klein-Gordon equations with Hulthen plus inversely quadratic exponential Mie-type potential,” Physical Science International Journal, vol. 19, no. 2, pp. 1–27, 2018.
View at: Publisher Site | Google Scholar
J. D. Chalk, “A study of barrier penetration in quantum mechanics,” American Journal of Physics, vol. 56, no. 1, pp. 29–32, 1988.
View at: Publisher Site | Google Scholar
E. M. Khokha, M. Abu-Shady, and T. A. Abdel-Karim, “Quarkonium Masses in the N-dimensional space using the exact analytical iteration method,” International Journal of Theoretical and Applied Mathematics, vol. 2, no. 2, pp. 86–92, 2016.
View at: Publisher Site | Google Scholar
S. M. Kuchin and N. V. Maksimenko, “Theoretical estimation of the spin-average mass spectra of heavy quarkonia and BC mesons,” Universal Journal of Physics and Application, vol. 1, no. 3, pp. 295–298, 2013.
View at: Publisher Site | Google Scholar
M. Abu-Shady, “Heavy Quarkonia and bc mesons in the Cornell potential with Harmonic oscillator potential in the N-dimensional Schrodinger equation,” International Journal of Applied Mathematics and Theoretical Physics, vol. 2, no. 2, pp. 16–20, 2016.
View at: Publisher Site | Google Scholar
A. Al-Jamel and H. Widyan, “Heavy quarkonium mass spectra in a coulomb field plus quadratic potential using Nikiforov-Uvarov method,” Applied Physics Research, vol. 4, no. 3, pp. 94–99, 2012.
View at: Publisher Site | Google Scholar
E. Omugbe, “Non-relativistic eigensolutions of molecular and heavy quarkonia interacting potentials via the Nikiforov Uvarov method,” Canadian Journal of Physics, vol. 98, no. 12, pp. 1125–1132, 2020.
View at: Publisher Site | Google Scholar
R. Bader and G. Jones, “The Hellmann-Feynman theorem and chemical binding,” Canadian Journal of Chemistry, vol. 39, no. 6, pp. 1253–1265, 1961.
View at: Publisher Site | Google Scholar
D. Cremer and E. Kraka, “A description of chemical bond interms of local properties of electron density and energy,” Croatica Chemica Acta, vol. 57, no. 6, pp. 1259–1281, 1984.
View at: Google Scholar
T. Berlin, “Binding regions in diatomic molecules,” The Journal of Chemical Physics, vol. 19, no. 2, pp. 208–213, 1951.
View at: Publisher Site | Google Scholar
P. Politzer and J. S. Murray, “The Hellmann-Feynman theorem: a perspective,” Journal of Molecular Modelling, vol. 24, no. 9, 2018.
View at: Publisher Site | Google Scholar
E. Wilson Jr., “Four‐dimensional electron density function,” The Journal of Chemical Physics, vol. 36, no. 8, pp. 2232-2233, 1962.
View at: Publisher Site | Google Scholar
I. B. Okon, O. Popoola, and C. N. Isonguyo, “Approximate solutions of Schrodinger equation with some diatomic molecular interactions using Nikiforov-Uvarov method,” Advances in High Energy Physics, vol. 2017, Article ID 9671816, 24 pages, 2017.
View at: Publisher Site | Google Scholar
I. B. Okon, A. D. Antia, L. E. Akpabio, and B. U. Archibong, “Expectation values of some diatomic molecules with Deng-Fan potential using Hellmann-Feynman theorem,” Journal of Applied Physical Science International, vol. 10, no. 5, pp. 247–268, 2019.
View at: Google Scholar
R. P. Feynman, “Forces in molecules,” Physical Review, vol. 56, no. 4, pp. 340–343, 1939.
View at: Publisher Site | Google Scholar
K. J. Oyewumi and K. D. Sen, “Exact solutions of the Schrödinger equation for the pseudoharmonic potential: an application to some diatomic molecules,” Journal of Mathematical Chemistry, vol. 50, no. 5, pp. 1039–1059, 2012.
View at: Publisher Site | Google Scholar
S. M. Ikhdair, “An improved approximation scheme for the centrifugal term and the Hulthén potential,” The European Physical Journal A, vol. 39, no. 3, pp. 307–314, 2009.
View at: Publisher Site | Google Scholar
I. B. Okon, C. N. Isonguyo, A. D. Antia, A. N. Ikot, and O. O. Popoola, “Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential,” Communications in Theoretical Physics, vol. 72, no. 6, 2020.
View at: Publisher Site | Google Scholar
J. Stanek, “Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term,” Central European Journal of Chemistry, vol. 9, no. 4, pp. 737–742, 2011.
View at: Publisher Site | Google Scholar
W. Lucha and F. F. Schöberl, “Solving the Schrödinger equation for bound states with Mathematica 3.0,” International Journal of Modern Physics C: Computational Physics and Physical Computation, vol. 10, no. 4, pp. 607–619, 1999.
View at: Publisher Site | Google Scholar
C.-S. Jia, J.-Y. Liu, and P.-Q. Wang, “A new approximation scheme for the centrifugal term and the Hulthen potential,” Physics Letters A, vol. 372, no. 27-28, pp. 4779–4782, 2008.
View at: Publisher Site | Google Scholar
H. Hassanabadi, E. Maghsoodi, A. N. Ikot, and S. Zarrinkamar, “Approximate arbitrary-state Solutions of Dirac equation for modified Hylleraas and modified Eckart potentials by NU method,” Applied Mathematics and computation, vol. 219, no. 17, pp. 9388–9398, 2013.
View at: Publisher Site | Google Scholar
E. E. Ituen, A. D. Antia, and I. B. Okon, “A nalytical solutions of non-relativistic Schrodinger equation with hadronic mixed power-law potentials via Nikiforov-Uvarov method,” World Journal of Applied Science and Technology, vol. 4, no. 2, pp. 196–205, 2012.
View at: Google Scholar
B. T. Mbadjoun, J. M. Ema’a, J. Yomi, P. E. Abiama, G. H. Ben-Bolie, and P. O. Ateba, “Factorization method for exact solution of the non-central modified Killingbeck potential plus a ring-shaped like potential,” Modern Physics Letters A, vol. 34, no. 10.1142/S021773231950072X, 2019.
View at: Publisher Site | Google Scholar
O. A. Awoga and A. N. Ikot, “Approximate solution of Schrödinger equation in D dimensions for inverted generalized hyperbolic potential,” Pramana, vol. 79, no. 3, pp. 345–356, 2012.
View at: Publisher Site | Google Scholar
E. Omugbe, O. E. Osafile, and I. B. Okon, “WKB energy expression for the radial Schrödinger equation with a generalized pseudoharmonic potential,” Asian Journal of Physical and Chemical Sciences, vol. 8, no. 2, pp. 13–20, 2020.
View at: Publisher Site | Google Scholar
E. Omugbe, “Non-relativistic energy spectrum of the Deng-Fan oscillator via the WKB approximation method,” Asian Journal of Physical and Chemical Sciences, vol. 8, no. 1, pp. 26–36, 2020.
View at: Publisher Site | Google Scholar
C. N. Isonguyo, I. B. Okon, A. N. Ikot, and H. Hassanabadi, “Solution of Klein -Gordon equation for some diatomic molecules with new generalized Morse-like potential using SUSYQM,” Bulletin of the Korean Chemical Society, vol. 35, no. 12, pp. 3443–3446, 2014.
View at: Publisher Site | Google Scholar
C. Gang, “Bound states for Dirac equation with Wood-Saxon potential,” Acta Physica Sinica, vol. 53, no. 3, pp. 680–683, 2004.
View at: Publisher Site | Google Scholar
G. Chen, Z. D. Chen, and Z. M. Lou, “Exact bound state solutions of the s-wave Klein-Gordon equation with the generalized Hulthen potential,” Physics Letters A, vol. 331, no. 6, pp. 374–377, 2004.
View at: Publisher Site | Google Scholar
C. S. Jia, J. W. Dai, L. H. Zhang, J. Y. Liu, and G. D. Zhang, “Molecular spinless energies of the modified Rosen-Morse potential energy model in higher spatial dimensions,” Chemical Physics Letters, vol. 619, pp. 54–60, 2015.
View at: Publisher Site | Google Scholar
C. S. Jia, J. W. Dai, L. H. Zhang, Y. J. Liu, and X. L. Peng, “Relativistic energies for diatomic molecule nucleus motions with the spin symmetry,” Physics Letters A, vol. 379, no. 3, pp. 137–142, 2015.
View at: Publisher Site | Google Scholar
C. S. Jia, J. Chen, L. Z. Yi, and S. R. Lin, “Equivalence of the deformed Rosen-Morse potential energy model and Tietz potential energy model,” Journal of Mathematical Chemistry, vol. 51, no. 8, pp. 2165–2172, 2013.
View at: Publisher Site | Google Scholar
W. C. Qiang and S. H. Dong, “The Manning–Rosen potential studied by a new approximate scheme to the centrifugal term,” Physica Scripta, vol. 79, no. 4, 2009.
View at: Publisher Site | Google Scholar
X. Y. Gu and S. H. Dong, “Energy spectrum of the Manning-Rosen potential including centrifugal term solved by exact and proper quantization rules,” Journal of Mathematical Chemistry, vol. 49, no. 9, pp. 2053–2062, 2011.
View at: Publisher Site | Google Scholar
Z. Q. Ma, A. Gonzalez-Cisneros, B. W. Xu, and S. H. Dong, “Energy spectrum of the trigonometric Rosen-Morse potential using an improved quantization rule,” Physics letters A, vol. 371, no. 3, pp. 180–184, 2007.
View at: Publisher Site | Google Scholar
H. Hassanabadi, S. Zarrinkamar, and A. A. Rajabi, “Exact solutions of D-dimensional Schrödinger equation for an energy-dependent potential by NU method,” Communication in Theoretical Physics, vol. 55, no. 4, pp. 541–544, 2011.
View at: Publisher Site | Google Scholar
H. Hassanabadi, E. Maghsodi, S. Zarrinkamar, and H. Rahimov, “An approximate solution of the Dirac equation for hyperbolic scalar and vector potentials and a coulomb tensor interaction by SUSYQM,” Modern Physics Letters A, vol. 26, no. 36, pp. 2703–2718, 2011.
View at: Publisher Site | Google Scholar
R. H. Parmar, “Solution of the ultra generalized exponential-hyperbolic potential in multi-dimensional space,” Few-Body Systems, vol. 61, no. 4, pp. 1–28, 2020.
View at: Publisher Site | Google Scholar
A. Diaf, A. Chouchaoui, and R. J. Lombard, “Feynman integral treatment of the Bargmann potential,” Annals of Physics, vol. 317, no. 2, pp. 354–365, 2005.
View at: Publisher Site | Google Scholar
C. A. Onate, “Bound state solutions of the SchrӦdinger equation with second PӦschl-Teller like potential model and the vibrational partition function, mean energy and mean free energy,” Chinese Journal of Physics, vol. 54, no. 2, pp. 165–174, 2016.
View at: Publisher Site | Google Scholar
I. B. Okon, O. O. Popoola, E. Omugbe, A. D. Antia, C. N. Isonguyo, and E. E. Ituen, “Thermodynamic properties and bound state solutions of Schrodinger equation with Mobius square plus screened-Kratzer potential using Nikiforov-Uvarov method,” Computational and Theoretical Chemistry, vol. 1196, 2021.
View at: Publisher Site | Google Scholar
E. Omugbe, O. E. Osafile, M. C. Onyeaju, I. B. Okon, and C. A. Onate, “The unified treatment on the non-relativistic bound state solutions, thermodynamic properties and expectation values of exponential–type potentials,” Canadian Journal of Physics, vol. 99, 2021.
View at: Publisher Site | Google Scholar
K. J. Oyewumi, B. J. Falaye, C. A. Onate, O. J. Oluwadare, and W. A. Yahya, “Thermodynamics properties and the approximate solutions of the Schrodinger equation with the shifted Deng-Fan potential model,” Molecular Physics, vol. 112, no. 1, pp. 127–141, 2014.
View at: Publisher Site | Google Scholar
A. N. Ikot, U. S. Okorie, R. Sever, and G. J. Rampho, “Eigensolution, expectation values and thermodynamic properties of the screened Kratzer potential,” European Physical Journal Plus, vol. 134, no. 8, 2019.
View at: Publisher Site | Google Scholar
U. S. Okorie, E. E. Ibekwe, A. N. Ikot, M. C. Onyeaju, and E. O. Chukwuocha, “Thermodynamic properties of the modified Yukawa potential,” Journal of the Korean Physical Society, vol. 73, no. 9, pp. 1211–1218, 2018.
View at: Publisher Site | Google Scholar
C. O. Edet, U. S. Okorie, A. T. Ngiangia, and A. N. Ikot, “Bound state solutions of the Schrodinger equation for the modified Kratzer potential plus screened Coulomb potential,” Indian Journal of Physics, vol. 94, no. 4, pp. 425–433, 2020.
View at: Publisher Site | Google Scholar
U. S. Okorie, A. N. Ikot, M. C. Onyeaju, and E. O. Chukwuocha, “A study of thermodynamic properties of quadratic exponential type potential in D-dimensions,” Revista Mexicana de fisica, vol. 64, no. 6, pp. 608–614, 2018.
View at: Publisher Site | Google Scholar
A. Boumali and H. Hassanabadi, “The Thermal properties of a two-dimensional Dirac oscillator under an external magnetic field,” The European Physical Journal Plus, vol. 128, no. 10, 2013.
View at: Publisher Site | Google Scholar
I. B. Okon, C. A. Onate, E. Omugbe et al., “Approximate solutions, thermal properties and superstatistics solutions to Schrödinger equation,” https://arxiv.org/abs/2110.09896.
View at: Google Scholar
O. Bayrak, G. Kocak, and I. Boztosun, “Anyl-state solutions of the Hulthén potential by the asymptotic iteration method,” Journal of Physics A: Mathematical and General, vol. 39, no. 37, pp. 11521–11529, 2006.
View at: Publisher Site | Google Scholar
S. M. Ikhdair and R. Sever, “Relativistic and nonrelativistic bound states of the isotonic oscillator by Nikiforov-Uvarov method,” Journal of Mathematical Physics, vol. 52, no. 12, 2011.
View at: Publisher Site | Google Scholar
H. Fakhri and J. Sadeghi, “Supersymmetry approaches to the bound states of the generalized Woods-Saxon potential,” Modern Physics Letters A, vol. 19, no. 8, pp. 615–625, 2004.
View at: Publisher Site | Google Scholar
H. Khounfais, T. Boudjedaa, and L. Chetouani, “Scattering matrix for Feshbach-Villars equation for spin 0 and 1/2: Woods-Saxon potential,” Czechoslovak Journal of Physics, vol. 54, no. 7, pp. 697–710, 2004.
View at: Publisher Site | Google Scholar
H. Hassanabadi, S. Zarrinkamar, H. Hamzavi, and A. A. Rajabi, “Exact solutions of D-dimensional Klein-Gordon equation with an energy-dependent potential by using of Nikiforov–Uvarov method,” The Arabian Journal for Science and Engineering, vol. 37, no. 1, pp. 209–215, 2012.
View at: Publisher Site | Google Scholar
A. Arda and R. Sever, “Approximate analytical solutions of a two-term diatomic molecular potential with centrifugal barrier,” Journal of Mathematical Chemistry, vol. 50, no. 7, pp. 1920–1930, 2012.
View at: Publisher Site | Google Scholar
S. M. Ikdair and R. Sever, “Exact solutions of the modified Kratzer potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the Nikiforov-Uvarov method,” International Journal of Modern Physics C, vol. 19, no. 2, pp. 221–235, 2008.
View at: Publisher Site | Google Scholar
C. S. Jia, P. Guo, Y. F. Diao, L. Z. Yi, and X. J. Xie, “Solutions of Dirac equations with the Pöschl-Teller potential,” The European Physical Journal A, vol. 34, no. 1, pp. 41–48, 2007.
View at: Publisher Site | Google Scholar
C. Tezcan and R. Sever, “A general approach for the exact solution of the Schrödinger equation,” International Journal of Theoretical Physics, vol. 48, no. 2, pp. 337–350, 2009.
View at: Publisher Site | Google Scholar
C. Beck, “Superstatistics: theory and applications,” Continuum Mechanics and Thermodynamics, vol. 16, no. 3, pp. 293–304, 2004.
View at: Publisher Site | Google Scholar
C. Beck and E. G. Cohen, “Superstatistics,” Physica A, vol. 322, pp. 267–275, 2003.
View at: Publisher Site | Google Scholar
C. O. Edet, P. O. Amadi, U. S. Okorie, A. Tas, A. N. Ikot, and G. Rampho, “Solutions of Schrodinger equation and thermal properties of generalized trigonometric Poschl-Teller potential,” Revista Mexicana defisca, vol. 66, pp. 824–839, 2020.
View at: Publisher Site | Google Scholar
H. Hamzavi, K. E. Thylwe, and A. A. Rajabi, “Approximate bound states solution of the Hellmann potential,” Communications in Theoretical Physics, vol. 60, no. 1, pp. 1–8, 2013.
View at: Publisher Site | Google Scholar
B. I. Ita, H. Lious, O. U. Akakuru et al., “Bound state solutions of the Schrödinger equation for the more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov-Uvarov method,” Journal of quantum Information Science, vol. 8, no. 1, pp. 24–45, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Ituen Okon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

PDF Download Citation

Download other formats

Order printed copies

Views

471

Downloads

678

Citations